32 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



The integration of these equations can, as already stated, be effected in the same 

 way as for the preceding part of the solution of the original equation ; the most 

 general solution of the system is found to be 



a 2f/ + c _ ,/,, \ 



15. That the method just expounded is not restricted to individual instances of 

 equations, for which A = is resoluble, can be seen as follows. 

 We consider, more generally, the case when 



Ap z + H/></ + B</ 2 Gp F<7 + C = 



is resoluble into two linear equations. We have 



(A/) + $Ilq - iG) 2 = (H* - AB) (f + (AF - GH) q + ,}G 2 - AC. 



Let 



1H- - AB = &\ 



AF - GH = - 20ty ; 

 then, since 



(III- - AB) (}G- - AC) = i (AF - iGH)-, 

 we have 



iG 3 - AC = 6'^, 

 and the equation is 



that is, we have the two equations 



(iH -e)q- (JG - ^) = o 



where 



6* = iH 2 - AB 

 Taking the former of the two equations, viz., 



we seek to obtain combinations of the three equations of identity with the three 

 equations particular to the present case. The first equations of each of these sets, as 

 given in 4, 5, are 



da dJt^ dg d<j 



dy ~ da; """ P rfy ~ ? ~dx ~ 



